Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(14,17)(18,20)(21,22,24,25)(23,26,28,27);(1,2,4,3)(6,8)(14,17)(18,20)(22,25)(26,27);(2,3)(5,6)(7,8)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,25)(26,28);(1,3,4,2)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de104 conjugacy classes — view the full character table.
| maximal subgroup of | #72331342, #72331344, #72331350, … |
| maximal normal subgroup of | #72331342, #72331344, #72331350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 62889);